A Degree Condition for Graphs Having All $(a,b)$-Parity Factors

TL;DR

This paper establishes a degree condition under which large graphs have all $(a,b)$-parity factors, expanding understanding of parity factors in graph theory with optimal bounds.

Contribution

It provides a new sufficient degree condition for graphs to have all $(a,b)$-parity factors, including bounds on minimum degree and maximum degree for nonadjacent vertices.

Findings

Graphs with sufficiently large minimum degree have all $(a,b)$-parity factors.

The vertex degree conditions are shown to be optimal in some cases.

The results apply to graphs with at least $3(b+1)(a+b)$ vertices.

Abstract

Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\ldots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 3(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.